How many paid bidders to recruit

You are selling gold at auction and will receive the second-highest bid. Each bidder's bid is independently uniform between $\text{\char36}500$ and $\text{\char36}1000$, and the auction house charges $\text{\char36}10$ for every bidder it recruits. If you recruit the number of bidders that maximises your expected net proceeds, what is that expected net amount?

$\text{\char36}810$ (with $9$ bidders). The second-highest of $n$ uniform order statistics on $[500,1000]$ sits on average at fraction $\tfrac{n-1}{n+1}$ of the range, i.e. $1000-\tfrac{1000}{n+1}$ dollars. Subtracting the $10n$ fee, the net is maximised at $n=9$: $900-90=810$ (versus about $808.9$ at $n=8$ and $809.1$ at $n=10$).